P.I.M. Teun Koetsier : Lakatos' Philosophy of Mathematics, A Historical Approach, Amsterdam, North-Holland, 1991 

Introduction

The present monograph deals with the methodology of mathematics. This does not, however, mean that the working mathematician should expect it to solve his (1) methodological problems. The study also deals with the history of mathematics, but here again the historiographer should not expect major contributions to the historiography of mathematics. This study is a contribution to the philosophy of mathematics and like most philosophical treatises it neither addresses itself to any practical problems, nor does it solve them.

In a paper published in 1981 the mathematician Mac Lane encouraged philosophers to renew the study of the philosophy of mathematics, a subject which he described as being "dormant since about 1931" (2) . Other authors have expressed themselves in a similar way. For example Putnam , in a paper published in 1979 on recent trends in the philosophy of mathematics, came to the conclusion that none of the existing views on the nature of mathematics were valid (3) . And in 1979, Goodman argued that the four major views in the philosophy of mathematics - formalism, intuitionism, logicism and platonism - arise from "an oversimplification of what happens when we do mathematics" (4) . From Goodman's point of view a more adequate philosophy of mathematics had yet to be formulated (5) . Tymoczko, editor of the 1986 anthology New Directions in the Philosophy of Mathematics (6) , strikes a more optimistic note in his introductions to the different parts of the book. From his point of view the papers collected in the anthology contain the outlines of a more adequate philosophy of mathematics which he calls quasi-empiricism. Tymoczko states: "This anthology delineates quasi-empiricism as a coherent and increasingly popular approach to the philosophy of mathematics" (7) .

What is quasi-empiricism ? Tymoczko uses a rather loose definition. For him quasi-empiricism is a philosophical position, or rather a set of related philosophical positions, that attempts to re-characterize the mathematical experience by taking the actual practice of mathematics seriously (8) . Tymoczko writes :

"If we look at mathematics without prejudice, many features will stand out as relevant that were ignored by the foundationalists: informal proofs, historical development, the possibility of mathematical error, mathematical explanations (in contrast to proofs), communication among mathematicians, the use of computers in modern mathematics, and many more" (9) .

Usually the word 'quasi-empiricism' is used in a more restricted sense. As far as I know, the word was first used by Lakatos in the middle of the sixties and also by Putnam later in the seventies. In 1967 Lakatos distinguished two different kinds of theories: quasi-empirical theories and Euclidean theories. He defined Euclidean theories as theories in which the characteristic truth flow inundating the whole system goes from the top, the axioms, down to the bottom. He defined quasi-empirical theories as theories in which the crucial truth flow is the upward transmission of falsity from the "basic statements" to the axioms. Attacking the foundationalist illusion that there exists a means of finding a foundation for mathematics which will be satisfactory once and for all, Lakatos argued that mathematics is not Euclidean, but instead quasi-empirical. Carried away by his own reasoning and wishing to show the fallibility of mathematics in the sense of Popper's falsificationism, Lakatos exaggerated. There is indeed an upward transmission of falsity in mathematics, but it is not the crucial truth flow. Yet Lakatos had a point. Later, in two papers published in 1975 and 1979, we also find Putnam defending the point of view that mathematics is quasi-empirical (10) . In the 1975 paper Putnam argued that:

"mathematical knowledge resembles empirical knowledge - that is, that the criterion of truth in mathematics just as much as in physics is success of our ideas in practice, and that mathematical knowledge is corrigible and not absolute" (11) .

In the 1979 paper Putnam presents his quasi-empirical realism as a modification of Quine'sholism. Roughly speaking, Quine's holism consists of the view that science as a whole is one comprehensive explanatory theory, justified by its ability to explain sensations. Mathematics and logic are part of this theory, differing from natural science in the sense that they assume a very central position. Since giving up logical or mathematical truths causes great upheaval in the network of our knowledge, they are not given up. Yet, in principle, mathematics and logic are no different from natural science (12) . Putnam's quasi-empirical realism of 1979 consisted of Quine's view, but with two modifications. First, Putnam added combinatorial facts (for example, the fact that a finite collection always receives the same count no matter in what order it is counted) to sensations as elements that mathematical theorems must explain. Secondly, Putnam required that there be agreement between mathematical theory and mathematical "intuitions" whatever their source (for example the self-evidence of the Comprehension Axioms in set theory). Both Lakatos and Putnam considered mathematical theories to be interrelated sets of statements that are considered to be true. The quasi-empirical element in their positions is the fact that they reject the view that, in principle, mathematics could be described in a Euclidean way in Lakatos's sense of the term. Both Lakatos and Putnam argue that, to a certain extent, mathematical theories always possess a hypothetical status. In that respect mathematical knowledge resembles empirical knowledge.

This study is an essay in quasi-empiricism in every sense of the word. It is an attempt to look at the history of mathematics from a Lakatosian point of view. The actual historical development of mathematics is taken very seriously in a way inspired by Lakatos's unfinished quasi-empiricist philosophical work on mathematics. The first two chapters are devoted to a critical discussion of Lakatos's views and in the remaining chapters the emphasis is on the development of a notion of mathematical rationality that agrees with the historical facts.

Lakatos's position can be summarized in the form of two theses:

I. Lakatos's fallibility thesis: Fallibility is an essential characteristic of mathematical knowledge. Most philosophies of mathematics are infallibilist. Infallibilists argue that, although in practice mathematicians make mistakes, mathematical knowledge is essentially infallible. In fact Lakatos's quasi-empiricism consists in the fallibility thesis. Although their subject matter is different, mathematical theories and empirical theories have in common the fact that they are fallible.

II. Lakatos's rationality thesis: In spite of its fallible character the development of mathematical research is not completely arbitrary, but possesses its own rationality. Fallible mathematical knowledge is replaced by other fallible knowledge in accordance with certain norms of rationality.

Most of Lakatos's work with respect to mathematics concentrates on the rationality thesis. A major tool that Lakatos uses is the rational reconstruction of developments in mathematics. A rational reconstruction is a reconstruction that is explicitly based on a particular methodology. In Lakatos's work one can distinguish two periods. During the first period he attempted to apply a Popperian brand of falsificationism to mathematics. In this period the Methodology of Proofs and Refutations (MP&R) dominated his work. In the second half of the sixties, when Lakatos developed his Methodology of Scientific Research Programmes (MSRP), the second period started. During this period he wrote primarily on natural science.

In Chapter I, I concentrate on two of Lakatos's rational reconstructions of historical developments in mathematics from the first period. The first reconstruction concerns the history of Euler's formula for polyhedra, and the second reconstruction concerns the history of the notion of uniform convergence. In order to create clarity about their status the reconstructions are confronted directly with the actual historical development to which they refer. One conclusion of Chapter I is that, although the rational reconstructions on the basis of the MP&R are quite counter-factual when taken as reconstructions of historical developments, they reveal important heuristic patterns. Another conclusion is that neither Lakatos's reconstructions nor the corresponding actual historical developments exhibit more than the weak fallibility of mathematical knowledge.

In Chapter II, I discuss Lakatos's paper "A Renaissance of Empiricism in the Recent Philosophy of Mathematics", written in the middle of the sixties which clearly shows the problems that Lakatos's early falsificationist views of mathematics encountered. Why is it so difficult to identify potential falsifiers in mathematics ? Why is the illusion that mathematics is an infallible science so strong ? Some of these problems were, in principle, solved by Lakatos's MSRP, which Lakatos considered applicable to mathematics although it was developed for natural science as an immediate reaction to Kuhn's work on the development of natural science. The conclusion of Chapter II is that the MSRP needs to be modified in order to be applicable to mathematics, and that it would be worthwhile to investigate the validity of the following picture of mathematical rationality which results from such a modification. On the micro-level (roughly speaking, the level of conjectures and theorems), the overall mathematical methodology is the MP&R (viz: conjectures, attempts at proofs, refutations, critical proof-analysis, etc.). Dramatic refutations, however, are rather rare, particularly when the mathematicians involved are cautious and do not prematurely consider plausible conjectures as theorems. On the micro-level, conjectures, theorems and proofs are fallible in the sense that they can be replaced by other conjectures, theorems and proofs along the lines of the MP&R. On the macro-level (roughly, the level of whole fields, or even mathematics as a whole) the situation is different. There, the developments can no longer be described in terms of the MP&R. The development of mathematics shows that, in practice, coherent clusters of considerable numbers of theorems become established. They are no longer subject to refutation, but form the core of a theory or a set of theories that is being developed further. Such developing theories or sets of theories can be called "research programmes". With respect to such "research programmes" one could also define a form of refutation. The refutation of a "research programme", however, is not a logical refutation but an expression of the fact that the programme has lost ground in competition with another programme. On the macro-level, research programmes are fallible in the sense that they can be replaced.

Chapters III through IX of the book are primarily devoted to an evaluation of the above picture of mathematical rationality. Only in the last chapter do I return briefly to the fallibility thesis. In Chapter III, I discuss Lakatos's paper "Cauchy and the Continuum: the Significance of Non-Standard Analysis for the History and the Philosophy of Mathematics". This paper is interesting because, although it was written a year before Lakatos introduced the notion of research programme, it in fact contains a reconstruction of two competing research programmes in mathematics. In Chapter IV, I discuss an attempt by Spalt (*) to develop these ideas further and to describe the history of analysis explicitly in terms of competing research programmes. From the failure of his own attempt at reconstruction, Spalt drew the conclusion that all attempts to characterize mathematical rationality in terms of the MSRP or related models are bound to fail. I am less pessimistic. There are circularity problems to be solved, and, in order to avoid vicious circularity, attempts to apply (a modified version of) the MSRP to mathematics require a careful "meta-methodological" approach. The outline of such an approach is also given in Chapter IV.

In Chapter V, I turn to two papers by Giorello (**) . A critical examination of these suggests that we have to modify Lakatos's MSRP as follows. Mathematicians always work on research projects that are characterized by their heuristic unity. Research projects usually belong to a research tradition. Research traditions concern a particular fundamental mathematical domain and are characterized by the entities that are being studied in that domain and by assumptions about the appropriate methods by which to prove the properties of those entities. A fundamental mathematical domain consists of the most general mathematical entities that still function mathematically in a certain period of time. A good example of a mathematical research tradition is the dominant deductive geometric tradition, represented, for example, by Euclid's Elements. I call that tradition the Euclidean Tradition. The fundamental domain of that tradition is Euclidean space. Another example of a research tradition is the Structuralist Tradition that dominated the twentieth century. I propose a Methodology of Mathematical Research Traditions (MMRT) in Chapter VI. I follow a suggestion by Marchi (***) and make mathematical theorems analogous to facts in natural science. The progress of research projects and research traditions is defined in terms of the weight of the conjectures and theorems that are being generated by a project or tradition. In Chapters VII and VIII, two rational reconstructions on the basis of the MMRT are given. The first reconstruction, in Chapter VII, shows that one can argue on the basis of the existing historiographical material that in Greek mathematics the Euclidean Tradition was preceded by another tradition - the Demonstrative Tradition. Although any reconstruction of the development of pre-Euclidean mathematics will be rather speculative owing to a lack of primary sources, my speculations indicate that one must, in mathematics, distinguish deductive traditions from non-deductive traditions. The Demonstrative Tradition was a non-deductive tradition which was superseded by a deductive tradition, namely the Euclidean Tradition. Unlike the Demonstrative Tradition, the Euclidean Tradition possesses an explicitly defined "hard core" (analogous to the hard core that Lakatos takes as one of the defining elements of a research programme in natural science) that was formed in a process of repeated proofs and refutations.

In Chapter VIII, I describe the transition from eighteenth century analysis to nineteenth century analysis in terms of the MMRP. In my reconstruction, eighteenth century analysis was dominated by a particular tradition - the Formalist Tradition - which, to a considerable extent, was non-deductive. That tradition was superseded by a deductive tradition - the Conceptual Tradition. The rational reconstructions in Chapters VII and VIII show that the MMRT, within the limits that inherently accompany such models, satisfactorily characterizes the rationality of mathematics.

In Chapter IX, the history of a particular theorem in analysis - the interchangeability theorem for partial differentiation - gives us a good view of the internal development of the two traditions that were described in Chapter VIII. It becomes clear that mathematical research traditions are not rigid, but that, on the contrary, they are subject to change with time.

In Chapter X, I draw some conclusions. Mathematics is fallible, but the rational reconstructions in Chapters I through IX show that its fallibility is very weak. It is remarkable that the question: "How should we explain this weak fallibility?" brings us back to the classical questions in the philosophy of mathematics. I argue briefly that a realist position with respect to mathematics most naturally explains the development of mathematics seen from the point of view of the MMRT.

 

Teun Koetsier. Vrije Universiteit Amsterdam
Faculteit der Wiskunde en Informatica,


Notes [ Pour retourner à l'appel de note, cliquez sur à la fin de la note ]

1 - When I refer to mathematicians in general the reader should read "he or she" and "his or her" wherever I write respectively "he" and "his".
2 - [Mac Lane, 1981] , p.462
3 - [Putnam, 1979]
4 - [Goodman, 1979] , p.540
5 - [Goodman, 1979] , p.550
6 - [Tymoczko, 1986]
7 - [Tymoczko, 1986] , p. xvi.
8 - Recently Philosophica devoted two volumes to issues in the recent philosophy of mathematics ( [Van Bendegem, 1988 & 1989] ). The different contributions clearly show that something interesting is going on in the field. The existing gap between the classical philosophy of mathematics and mathematical praxis obviously has led to new perspectives on the problems in the philosophy of mathematics.
9 - [Tymoczko, 1986] , p.xvi
10 - [Putnam, 1975] and [Putnam, 1979]
11 - [Putnam, 1975] , p. 51; the emphasis is Putnam's.
12 - Quine defended this view of mathematics and logic on numerous occasions, for example in his well-known paper from 1951 entitled "Two dogmas of empiricism" (Reprinted in [Quine, 1953] ).


Literature

[Asquith & Hacking, 1981]: P. D. Asquith, I. Hacking (eds), Proceedings of the 1978 Biennal Meeting of the Philosophy of Science Association, Vol. II, East Lansing, Michigan, 1981

[Asquith & Kyburg, 1979]: P. D. Asquith, Henry E. Kyburg (eds), Jr., Current Research in Philosophy of Science, East Lansing Michigan, 1979

[Van Bendegem, 1988]: J. P. Van Bendegem, Non-Formal Properties of Real Mathematical Proofs, In PSA 1988, Proceedings of the 1988 Biennial Meeting of the Philosophy of Science Association Volume One, (A. Fine, J. Leplin eds), East Lansing, Michigan, 1988, pp. 249-254

[Van Bendegem, 1988]: J. P. Van Bendegem (editor), Philosophica 42, Recent Issues in the Philosophy of Mathematics I, 1988

[Van Bendegem, 1989]: J. P. Van Bendegem (editor), Philosophica 43, Recent Issues in the Philosophy of Mathematics II, 1989

[Cohen e.a., 1976]: R.S. Cohen, P.K. Feyerabend, M.W.Wartofsky (eds), Essays in Memory of Imre Lakatos, D. Reidel Publishing Company, Dordrecht-Holland / Boston-U.S.A., 1976

[Giorello, 1975]: G. Giorello, Archimedes and the Methodology of Research Programmes, Scientia 110, 1975, pp. 125-135 (Translation of G. Giorello, Archimede e la metodologia dei programmi di ricerca, Scientia 110, 1975, pp. 111-123

[Giorello, 1981]: G. Giorello, Intuition and Rigor: Some Problems of a 'Logic of Discovery' in Mathematics, pp. 113-135 in Maria Luisa Dalla Chiara (ed.), Italian Studies in the Philosophy of Science, Boston Studies in the Philosophy of Science, Vol. 47, Dordrecht, 1981

[Glas, 1986]: E. Glas, On the Dynamics of Mathematical Change in the Case of Monge and the French Revolution, Studies in the History and Philosophy of Science 17, 1986, pp. 249-268

[Glas, 1989 I]: E. Glas, Testing the Philosophy of Mathematics in the History of Mathematics, Part I: The Sociocognitive Process of Conceptual Change, Studies in the History and Philosophy of Science 20, 1989, pp. 115-131

[Glas, 1989 II]: E. Glas, Testing the Philosophy of Mathematics in the History of Mathematics, Part II: The Similarity between Mathematical and Scientific Growth, Studies in the History and Philosophy of Science 20, 1989, pp. 157-174

[Goodman, 1979]: N. D. Goodman, Mathematics as an Objective Science, American Mathematical Monthly 86, 1979, pp. 540-551

[Goodman, 1981]: N. D. Goodman, The Experiential Foundations of Mathematical Knowledge, History and Philosophy of Logic 2, 1981, pp. 55-65

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[Hallett, 1979]: M. F. Hallett, Towards a Theory of Mathematical Research Programmes , British Journal for the Philosophy of Science 30, 1979, pp. 1-25 and pp. 135-159 (This paper is based on parts of [Hallett,1979a])

[Hallett, 1979a]: M. F. Hallett, The Nature of Progress in Mathematics: Case studies in the Early History of Set Theory and topology, Ph. D. Thesis 1979, University of London

[ Mac Lane, 1981 ]: S. Mac Lane, Mathematical Models: a Sketch for the Philosophy of Mathematics, American Mathematical Monthly 88, 1981, pp. 462-472.

[Marchi, 1976]: P. Marchi, Mathematics as a Critical Enterprise, in [Cohen e.a.,1976], pp. 379-393

[Putnam, 1975]: H. Putnam, What is Mathematical Truth ?, In [Tymoczko,1986], pp. 49 -65 (Reprinted from H. Putnam, Philosophical Papers, Vol. 1, Cambridge: Cambridge University Press, 1975)

[Putnam, 1979]: H. Putnam, Philosophy of Mathematics: A Report, In [Asquith & Kyburg, 1979], pp. 386-398

[Quine, 1953]: W. V. Quine, From a Logical Point of View, Cambridge: Harvard, 1953

[Spalt, 1981]: D. D. Spalt, Vom Mythos der Mathematischen Vernunft, Darmstadt, 1981

[Tymoczko, 1986]. New directions in the philosophy of mathematics. 1986. Birkhaüser.
[Une nouvelle édition de cette anthologie est en préparation pour la fin de l'année (Princeton University Press)]

© North-Holland, 1991 [Merci à Teun]


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